Question

Find an angle θ with 0 ∘ < θ < 360 ∘ that has the same: Sine function value as 230 ∘ θ = _____ degrees Cosine function value as 230 ∘ θ = _____degrees

Answer 1

An angle with the same sine function value as 230° is 30.1° or 148.7°. An angle with the same cosine function value as 230° is 220°.

Step-by-step explanation:

To find an angle with the same sine function value as 230°, we need to find an angle whose sine equals the sine of 230°. We can use the unit circle to determine this. Since sine is positive in both the first and second quadrants, we need to find angles in these quadrants that have the same sine value as 230°. These angles are 30.1° and 148.7°.

To find an angle with the same cosine function value as 230°, we can use the complementary angle property of cosine. The cosine of an angle is equal to the sine of its complementary angle. So, an angle with the same cosine value as 230° would have a sine value of the complementary angle of 230°. The complementary angle to 230° is 90° - 230° = -140°. So, an angle with the same cosine value as 230° is -140°. However, since we are looking for an angle between 0° and 360°, we can add 360° to -140° to get an angle of 220°.

Answer 2

An angle θ between 0° and 360° that has the same sine value as 230° is 130°, and the angle with the same cosine value as 230° is 310°.

Step-by-step explanation:

The student is seeking to find an angle θ between 0° and 360° that has the same sine and cosine values as 230°.

To find the angle with the same sine value as 230°, we look for an angle in the first or second quadrant (since sine is positive for these quadrants) with the same reference angle as 230°. The reference angle for 230° is 50° (since 230° is in the third quadrant, and 180° + 50° = 230°). Therefore, the angle with the same sine value is 180° - 50°, which equals 130°.

For the cosine value, we need to find the angle in the fourth quadrant since cosine is negative in the second and third quadrants and the cosine of 230° is negative. The reference angle is again 50°. Therefore, the angle with the same cosine value is 360° - 50°, which is 310°.